TITLE: Examples of absolutely continuous functions that are not Lipschitz.
QUESTION [26 upvotes]: I have just solved an exercise, which asked to show that function $f$ is Lipschitz implies that $f$ is absolutely continuous. However, I'm wondering if the converse is true. I can't seem to think of any counterexamples at the moment. I think I'm brain dead or something, so I could use some help.

REPLY [2 votes]: See we know that indefinite integral of an integrable function is absolutely continous and these are the only absolutely continous functions.
So we want to find such f which is integrable but unbounded then its indefinite integral F(say) will be absolutely continous but derivative of F which is precisely f a.e. is unbounded and hence F is not lipschitz function.
we can define $f:(0,1)\to \mathbb R$ as
$f(x)=(x)^\frac{-1}{2} = \frac{1}{\sqrt{x}}$
then clearly $f$ is unbounded 
but it is an integrable function as its integral is $2$ on $(0,1)$ which is finite.
So its indefinite integral which is $2\sqrt(x)$ is our required absolutely continous function which is not lipschitz(as its derivative $f$ is unbounded)